Determination method for a position signal

ABSTRACT

Two sensors scan a measuring scale, which can be displaced in relation to the sensors and comprises a plurality of equidistant measuring gradation, and deliver corresponding measuring signals. The measuring signals are periodic during a uniform relative displacement of the measuring scale, essentially sinusoidal and essentially phase-shifted by 90° in relation to one another. They have an essentially identical amplitude and a base frequency that corresponds to the relative displacement of the measuring scale. During a delivery period of measuring signals, the measuring scale carries out a relative displacement through one measuring gradation. Corrected signals are determined from the measuring signals using correction values. A signal of the position of the measuring scale in relation to the sensors is determined in turn using said correction signals. Fourier coefficients are determined in relation to the base frequency for the corrected signals or for at least one supplementary signal that is derived from the corrected signals, said coefficients being used in turn to update the correction values. Said correction values contain two shift correction values at least one amplitude correction value and at least one phase correction value for the measuring signals, or part of said values, in addition to at least one correction value for at least one higher frequency wave of the measuring signals.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is the US National Stage of International Application No. PCT/EP2005/053681, filed Jul. 28, 2005 and claims the benefit thereof. The International Application claims the benefits of German application No. 10 2004 038 621.8 DE filed Aug. 9, 2004, both of the applications are incorporated by reference herein in their entirety.

FIELD OF INVENTION

The present invention relates to a determination method for a position signal, with which

-   -   two sensors scan a measuring scale which is moveable relative to         the sensors, and has a plurality of equally-spaced scale         divisions and thereby supply corresponding measuring signals,     -   for a uniform relative movement of the measuring scale the         measuring signals are periodic, are essentially sinusoidal, have         essentially the same amplitude, have a phase offset relative to         one another which is essentially 90°, have a basic frequency         which corresponds essentially with the relative movement of the         measuring scale, and over the course of one period of the         measuring signals the measuring scale executes a relative         movement of one scale division,     -   by applying correction values, corrected signals are determined         from the measuring signals,     -   using the corrected signals, a position signal of the measuring         scale is determined relative to the sensors,     -   for the corrected signals, or for at least one supplementary         signal derived from the corrected signals, Fourier coefficients         are determined relative to the basic frequency,     -   the correction values are adjusted by reference to the Fourier         coefficients,     -   the correction values include two offset correction values, at         least one amplitude correction value and at least one phase         correction value for the measuring signals or for some of these         values.

BACKGROUND OF INVENTION

Determination methods of this type are used in so-called incremental position sensors. With them, the measuring signals are generally referred to as the cosine and sine signals. By evaluation of the passages through zero of the measuring signals, a coarse position is determined—to an accuracy of one signal period. By evaluating in addition the values of the cosine and sine signals themselves, it is possible to determine—within a signal period, a fine position. For ideal measuring signals x, y, this then gives the position signal φ within the signal period concerned as φ=arctan(y/x) if x>0  (1) φ=arctan(y/x)+π if x<0  (2) φ=π/2 sign(y) if x=0  (3)

In practice, however, the measuring signals x, y are not ideal, but subject to error. With the state of the art, the formulation most commonly adopted for the erroneous measuring signals x, y is x=a cos(φ+Δ)+x ₀  (4) y=(1+m)a sin(φ)=y ₀  (5)

Here, x₀ and y₀ are offset errors in the measuring signals x and y, m is an amplitude error and Δ is a phase error. a is a signal amplitude. Methods for determining and compensating for these error quantities are generally known.

Thus, for example, a determination method of the type mentioned in the introduction is known from DE-A-101 63 504.

Determination methods for a position signal are known, from DE-A-100 34 733, from DE-A-101 63 528 and from the technical article “Erhöhung der Genauigkeit bei Wegsystemen durch selbstlemende Kompensation systematischer Fehler” [Increasing the precision of position measuring systems by self-learning compensation of systematic errors] by B. Höscheler, conference volume on SPS[PLC]/IPC/DRIVES, Elektrische Automatisierungstechnik—Systeme und Komponenten, Fachmesse und Kongress [Electrical Automation Technology—Systems and Components, Technical Fair and Congress] 23^(rd)-25^(th) Nov. 1999, Nuremberg, pages 617 to 626, by which:

-   -   two sensors scan a measuring scale, which is moveable relative         to the sensors and has a plurality of equally-spaced scale         divisions, and thereby supply corresponding measuring signals,     -   for a uniform relative movement of the measuring scale the         measuring signals are periodic, are essentially sinusoidal, have         essentially the same amplitude, have a phase offset relative to         one another which is essentially 90°, have a basic frequency         which corresponds essentially with the relative movement of the         measuring scale, and over the course of one period of the         measuring signals the measuring scale executes a relative         movement of one scale division,     -   by applying correction values, corrected signals are determined         from the measuring signals,     -   using the corrected signals, a position signal of the measuring         scale is determined relative to the sensors,     -   the correction values are adjusted,     -   the correction values include two offset correction values, at         least one amplitude correction value and at least one phase         correction value for the measuring signals.

With the determination method according to DE-A-100 34 733 and the technical article by B. Hoscheler, the position signal is then post-corrected by means of a fine correction method, to compensate for residual errors due to harmonics in the measuring signals. However, the fine correction method described there only works satisfactorily if any changes in speed which occur are sufficiently small.

EP-A-1 046 884 discloses a method for determining a position signal with which two sensors scan a measuring scale, which is moveable relative to the sensors and has a plurality of equally-spaced scale divisions, and thereby supply corresponding measuring signals. For a uniform relative movement of the measuring scale the measuring signals are periodic, have essentially the same amplitude, are essentially sinusoidal, have a phase offset relative to one another which is essentially 90°, and have a basic frequency which corresponds essentially with the relative movement of the measuring scale. Over the course of one period of the measuring signals the measuring scale executes a relative movement of one scale division. The measuring signals are detected with a time displacement relative to one another. For one of the measuring signals, a corrected signal is determined from the measuring signals, using correction values. A position signal of the measuring scale is determined relative to the sensors by reference to the corrected signal and the other, uncorrected signal.

SUMMARY OF INVENTION

An object of the present invention consists in specifying a method which is as simple as possible to carry out, giving as complete a correction as possible of the errors contained in the measuring signals, and which also works properly for larger speed changes.

This object is achieved by a determination method described herein.

The embodiment in accordance with the invention can be further simplified by determining the correction values only for those higher-frequency waves in the measuring signal, the frequency of which is an odd number multiple of the basic frequency. The components which are an even number multiple of the basic frequency are in many cases negligibly small.

The determination method in accordance with the invention can be yet further simplified if the correction values are determined only for those higher-frequency waves in the measuring signal, the frequency of which is three or five times the basic frequency, and the correction values for the higher-frequency waves in the measuring signal, the frequency of which is five times that of the basic frequency, have a predetermined ratio to the correction values for the higher-frequency waves in the measuring signal, the frequency of which is three times that of the basic frequency. In particular it is even possible to determine only the correction values for those higher-frequency waves in the measuring signals, the frequency of which is three times that of the basic frequency, so that the ratio is zero.

Implementation of the determination method in accordance with the invention is particularly simple if

-   -   for the purpose of determining the Fourier coefficients the         corrected signals, or the at least one supplementary signal as         applicable, are saved into one of several registers,     -   an angular range is assigned to each of the registers,     -   in each case the save is made into the register which has the         angular range within which the arctangent for the corrected         signal lies, and     -   the Fourier coefficients are determined by reference to the         values saved into the registers.

If the values stored in the registers are deleted after the Fourier coefficients have been determined, and a new determination of the Fourier coefficients is only then undertaken after the resisters have been adequately filled, this results in a particularly robust determination of the correction values.

In the ideal case, the registers are only adequately filled when values have been saved into all the registers in accordance with the method described above. However, it is also possible to regard the registers as adequately filled at the point when values have been saved into a first group of the registers in accordance with the method described above, and in this case a second group of the registers is filled with values which are determined by reference to the values saved in accordance with the method described above.

The evaluation of the values saved into the registers is particularly simple if certain registers are assigned to each Fourier coefficient and the Fourier coefficient concerned is determined by reference exclusively to the values which are saved in the registers assigned to the Fourier coefficient concerned. For appropriate assignment of the registers to the Fourier coefficients, it is then even possible to determine the Fourier coefficients solely by the formation of sums and differences of the values saved in the assigned registers.

The correction of the measuring signals is particularly optimal if

-   -   for the purpose of determining the corrected signals         pre-corrected signals are first determined,     -   the pre-corrected signals are determined from the measuring         signals, with reference to the offset correction values, to the         at least on amplitude correction values and/or to the at least         one phase correction value, and     -   the corrected signals are then determined by reference to the         pre-corrected signals and to the at least one correction value         for the one or more higher-frequency wave in the measuring         signals.

Various approaches are possible for determining the corrected signals from the pre-corrected signals. It is thus possible, for example, to first determine a preliminary arctangent from the pre-corrected signals, and then to determine the corrected signals by applying the preliminary arctangent as the argument in a Fourier series expansion.

It is thus possible to determine the corrected signals by reference to the pre-corrected signals, by the formation of functions of the form

$\begin{matrix} {x_{\propto} = {x_{c} - {a{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}{\cos\left( {q\;\varphi_{c}} \right)}} + {d_{q}{\sin\left( {q\;\varphi_{c}} \right)}}} \right\rbrack}}}} & (6) \\ {and} & \; \\ {y_{\propto} = {y_{c} - {a{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}{\cos\left( {{q\;\varphi_{c}} - {q\;{\pi/2}}} \right)}} + {d_{q}{\sin\left( {{q\;\varphi_{c}} - {q\;{\pi/2}}} \right)}}} \right\rbrack}}}} & (7) \end{matrix}$ where x_(cc) and y_(cc) are the corrected signals, x_(c) and y_(c) the pre-corrected signals, a the signal amplitude, c_(q) and d_(q) are weighting factors determined by reference to the Fourier coefficients and φ_(c) the preliminary arctangent.

The approach just described can be simplified by replacing the expression cos (qφ_(c)−qπ/2) by cos(qφ _(c)) for q=0, 4, 8,  (8) by sin(qφ_(c)) for q=1, 5, 9,  (9) by −cos(qφ _(c)) for q=2, 6, 10, . . . and  (10) by −sin(qφ _(c)) for q=3, 7, 11,  (11) and the expression sin(qφ_(c)−qπ/2) by sin(qφ _(c)) for q=0, 4, 8,  (12) by −cos(qφ _(c)) for q=1, 5, 9,  (13) by −sin(qφ _(c)) for q=2, 6, 10, . . . and  (14) by cos(qφ _(c)) for q=3, 7, 11,  (15)

It is possible to effect further simplification by replacing the expression cos (qφ_(c)) by the expression

$\begin{matrix} {\sum\limits_{r = 0}^{{int}{({q/2})}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {2\; r} \end{pmatrix}\left( {\cos\;\varphi_{c}} \right)^{q - {2\; r}}\left( {\sin\;\varphi_{c}} \right)^{2\; r}}} & (16) \end{matrix}$ and the expression sin(qφ_(c)) by the expression

$\begin{matrix} {\sum\limits_{r = 0}^{{int}{\lbrack{{({q - 1})}/2}\rbrack}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {{2r} + 1} \end{pmatrix}\left( {\cos\;\varphi_{c}} \right)^{q - {2r} - 1}\left( {\sin\;\varphi_{c}} \right)^{{2\; r} + 1}}} & (17) \end{matrix}$

It is even possible to avoid the determination of trigonometric function values, if the expression cos(qφ_(c)) is finally replaced by the expression x_(c)/a and the expression sin(qφ_(c)) by the expression y_(c)/a.

An alternative possibility consists in determining the corrected signals by reference to the pre-corrected signals, by forming functions of the form

$\begin{matrix} {x_{cc} = {x_{c} - {\sum\limits_{q = 2}^{\infty}{b_{q}x_{c}^{q}}}}} & (18) \\ {and} & \; \\ {y_{cc} = {y_{c} - {\sum\limits_{q = 2}^{\infty}{b_{q}y_{c}^{q}}}}} & (19) \end{matrix}$ where x_(cc) and y_(cc) are the corrected signals and x_(c) and y_(c) the pre-corrected signals, and b_(q) a weighting factor.

BRIEF DESCRIPTION OF THE DRAWINGS

Further advantages and details are given by the following description of an exemplary embodiment, in conjunction with the drawings. These show, as schematic diagrams:

FIG. 1 a block diagram of a determination circuit for a position signal,

FIG. 2 a first form of embodiment of a first extract from FIG. 1,

FIG. 3 a first form of embodiment of a second extract from von FIG. 1,

FIG. 4 a simplification of the approach in FIG. 3,

FIG. 5 a simplification of the approach in FIG. 4,

FIG. 6 a simplification of the approach in FIG. 5,

FIG. 7 a second form of embodiment of the first extract from FIG. 1,

FIG. 8 another extract from the determination circuit in von FIG. 1,

FIG. 9 an assignment of angular ranges to registers,

FIG. 10 a logical combination,

FIG. 11 a first approach for determining the correction values,

FIG. 12 an alternative approach for determining the correction values; and

FIG. 13 correction values for the higher-frequency waves in the measuring signals adjusted using Fourier coefficients.

DETAILED DESCRIPTION OF INVENTION

As shown in FIG. 1, a determination circuit, by means of which a position signal φ_(cc) is to be determined, has two sensors 1, 2 and a measuring scale 3. The measuring scale 3 is moveable relative to the sensors 1, 2. As shown in FIG. 1, it can for example be rotated about an axis of rotation 4. This is indicated in FIG. 1 by an arrow A. The measuring scale 3 has numerous (e.g. 1000 to 5000) equally-spaced scale divisions 5. The sensors 1, 2 scan the measuring scale 3 and thereby supply corresponding measuring signals x, y.

In the ideal case, the sensors 1, 2 have exactly equal sensitivities, and are ideally positioned. For a uniform movement of the measuring scale 3 relative to the sensors 1, 2, the latter are therefore in a position to supply measuring signals x, y which satisfy the following conditions:

-   -   They are periodic.     -   They have an equal amplitude.     -   They are exactly sinusoidal.     -   They have a phase offset relative to each other of exactly 90°.     -   They have a basic frequency fG which corresponds to the relative         movement of the measuring scale 3.

One period of the measuring signals x, y then corresponds to a relative movement of one scale division 5 by the measuring scale 3.

In the ideal case therefore, the following applies within one scale division 5: x=a cos(φ)  (20) y=a sin(φ)  (21) where a is the amplitude of the measuring signals x, y. Correspondingly, the following applies for the position signal φ of the measuring scale 3 within a scale division 5: φ=arctan(y/x) when x>0  (22) φ=arctan(y/x)+π when x<0  (23) φ=(π/2)sign(y) when x=0  (24)

In a real situation however, the sensors 1, 2 are not exactly positioned and they also have sensitivities which are—at least slightly—different. In the real situation therefore, for a uniform relative movement of the measuring scale 3 the measuring signals x, y have amplitudes which are only broadly the same, are only broadly sinusoidal in shape and only broadly have a phase offset of 90° relative to each other. On the other hand, the basic frequency fG of the measuring signals x, y is retained.

The following formulation can therefore be made for the measuring signals x, y as a function of the actual position φ of the measuring scale 3 within a scale division 5: x=a c(φ+Δ)+x ₀  (25) y=(1+m) a s(φ)+y ₀  (26) where c and s are periodic functions of the form c(φ)=cos(φ)+Σ_(q=2) ^(∞) [c _(q) cos(qφ)+d _(q) sin(qφ)]  (27) and s(φ)=sin(φ)+Σ_(q=2) ^(∞) [c _(q) cos(qφ−qπ/2)+d _(q) sin(qφ−qπ/2)]  (28)

The functions c and s are phase-shifted relative to one another by 90° or π/2, as applicable. Hence s(φ)=c(φ−π/2) applies.

In the above formulae, x₀ and y₀ represent offset errors, m an amplitude error and Δ a phase error. c_(q) and d_(q) are tracking signal distortions due to harmonics of the basic frequency fG, that is distortions caused by higher-frequency waves in the measuring signals x, y. The following applies as a general rule |x ₀ /a|, |y ₀ /a|, |m|, |2Δ/π|, |c _(q) |, |d _(q)|<<1.  (29)

These signal errors must be determined and compensated for.

The method in accordance with the invention is executed iteratively. It is now assumed below that values have already been determined for the signal errors x₀, y₀, m, Δ, c_(q), d_(q). At the start of the method, however, the value can be set to predetermined starting values, e.g. to x₀=y₀=m=Δ=c_(q)=d_(q)=0.

The measuring signals x, y detected by the sensors 1, 2 are initially fed to a first correction block 6, as shown in FIG. 1. Also fed to the correction block 6 are the correction values x₀, y₀, m and Δ for the offset, amplitude and phase errors. The first correction block 6 determines from these—see FIG. 2—pre-corrected signals x_(c), y_(c) in accordance with the ratios y _(c)=(y−y ₀)/(1+m)  (30) x _(c)=(x−x ₀ +y _(c) sin Δ)/cos(Δ)  (31)

For the pre-corrected signals x_(c), y_(c), the following approximations apply x _(c) ≈a cos(φ)+aΣ _(q=2) ^(∞) [c _(q) cos(qφ)+d _(q) sin(qφ)]  (32) y _(c) ≈a sin(φ)+aΣ _(q=2) ^(∞) [c _(q) cos(qφ−qπ/2)+d _(q) sin(qφ−qπ/2)]  (33)

Using the pre-corrected signals x_(c), y_(c) and the correction values c_(q), d_(q) for the higher-frequency waves in the measuring signals x, y, it is then possible in a second correction block 7 to determine corrected signals x_(cc), y_(cc), in doing which the tracking signal distortions are also largely compensated.

There are several possibilities for determining the corrected signals x_(cc), y_(cc).

For example, using the pre-corrected signals x_(c), y_(c) it is possible—see FIG. 3—to determine first a preliminary arctangent φ_(c) from the ratios φ_(c)=arctan(y _(c) /x _(c)) when x _(c)>0  (34) φ_(c)=arctan(y _(c) /x _(c))+π when x _(c)<0  (35) φ_(c)=(π/2)sign(y _(c)) when x _(c)=0  (36) and thence to determine the corrected signals x_(cc), y_(cc) by utilizing the preliminary arctangent φ_(c) as the argument in a Fourier series expansion. The corrected signals x_(cc), y_(cc) are then formed in this case, for example, by forming functions of the form x _(cc) =x _(c) −aΣ _(q=2) ^(∞) [c _(q) cos(qφ _(c))+d _(q) sin(qφ _(c))]  (37) y _(cc) =y _(c) −aΣ _(q=2) ^(∞) [c _(q) cos(qφ _(c) −qπ/2)+d _(q) sin(qφ _(c) −qπ/2)]  (38)

For the corrected signals x_(cc), y_(cc) determined in this way, it is then true to a very good approximation that x _(cc) =a cos(φ  (39) y _(cc) =a sin(φ  (40)

By analogy with the formulae 1 to 3 it is thus possible, using the measuring signals x, y and the correction values x₀, y₀, m, Δ, c_(q), d_(q), to determine with great accuracy an arctangent φ_(cc), and hence also the position φ_(cc) of the measuring scale 3 within a scale division 5. That is to say, using the corrected signals x_(cc), y_(cc) it is possible to determine the position signal φ_(cc) for the measuring scale 3 relative to the sensors 1, 2, by using the equations φ_(cc)=arctan(y _(cc) /x _(cc)) when x _(cc)>0  (41) φ_(cc)=arctan(y _(cc) /x _(cc))+π when x _(cc)<0  (42) φ_(cc)=(π/2)sign(y _(cc)) when x _(cc)=0  (43)

It should be remarked at this point that for the purpose of determining the complete position of the measuring scale 3 it is also necessary to know which scale graduation 5 has just been sensed by the sensors 1, 2 (the so-called coarse position). However, it is generally known how to determine this coarse position, and this is not a subject of the present invention. Rather, within the context of the present invention it is taken as given.

Formulae 37 and 38 are mathematically correct, but require a large computational effort because sine and cosine values must be determined both for qφ_(c) and for (qφ_(c)−qπ/2). For this reason, in accordance with the generally familiar addition theorems for sine and cosine the following substitutions—see FIG. 4—are made:

The expression cos(qφ_(c)qπ/2) is replaced by cos(qφ _(c)) for q=0, 4, 8,  (44) by sin(qφ _(c)) for q=1, 5, 9,  (45) by −cos(qφ _(c)) for q=2, 6, 10, . . . and  (46) by −sin(qφ _(c)) for q=3, 7, 11,  (47)

In addition, the expression sin(qφ_(c)−qπ/2) is replaced by sin(qφ _(c)) for q=0, 4, 8,  (48) by −cos(qφ _(c)) for q=1, 5, 9,  (49) by −sin(qφ _(c)) for q=2, 6, 10, . . . and  (50) by cos(qφ _(c)) for q=3, 7, 11,  (51)

After this it only remains necessary to determine the sine and cosine values of qφ_(c).

Formula 37, and the modified formula 38 which is arrived at by modification in accordance with the formulae 44 to 51, can however be yet further simplified. Because it is possible, as shown in FIG. 5, to replace the expression cos(qφ_(c)) in these formulae by the expression

$\begin{matrix} {\sum\limits_{r = 0}^{{int}{({q/2})}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {2\; r} \end{pmatrix}\left( {\cos\;\varphi_{c}} \right)^{q - {2\; r}}\left( {\sin\;\varphi_{c}} \right)^{2\; r}}} & (52) \end{matrix}$

Furthermore, the expression sin(qφ_(c)) can be replaced by the expression

$\begin{matrix} {\sum\limits_{r = 0}^{{int}{\lbrack{{({q - 1})}/2}\rbrack}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {{2\; r} + 1} \end{pmatrix}\left( {\cos\;\varphi_{c}} \right)^{q - {2\; r} - 1}\left( {\sin\;\varphi_{c}} \right)^{{2\; r} + 1}}} & (53) \end{matrix}$

After this it only remains necessary to determine the sine and cosine of φ_(c).

However, even the determination of these trigonometric functions can be avoided. Because it is possible—see FIG. 6—to replace the expression cos(φ_(c)) by the expression x_(c)/a and the expression sin(φ_(c)) by the expression y_(c)/a.

In many case, the measuring signals x, y arise from a mapping of the signals x _(cos) =a cos(φ+Δ)+x ₀  (54) y _(cos)=(1+m) a sin(φ)+y ₀  (55) by means of a (common) non-linear characteristic curve f. The following then applies x=f(x _(cos))  (56) y=f(y _(cos))  (57)

In this case, the correction values d_(q) vanish, that is they have a value of zero. In this case it is therefore possible—see FIG. 7—to determine the corrected signals x_(cc), y_(cc) using the pre-corrected signals x_(c), y_(c) by forming functions of the form x _(cc) =x _(c−Σ) _(q=2) ^(∞) b _(q) x _(c) ^(q)  (58) y _(cc) =y _(c−Σ) _(q=2) ^(∞) b _(q) y _(c) ^(q)  (59)

Here, the coefficients bq are determined by the ratio b_(q)=a⁻¹Σ_(q′=q) ^(Q)h_(q,q′)c_(q)  (60) where h_(q,q′) are matrix coefficients. Here, the matrix coefficients h_(q,q′) can be determined as follows:

For the sake of simplicity and with no loss of generality, the assumption is initially made in what follows that the correction values x₀, y₀, m and Δ are zero.

We now assume further that the non-linear function f can be expanded as a Taylor series and the Taylor coefficients of the function f correspond to the coefficients b_(q) and that |bq|<<1. Then the measuring signal x resulting from a position φ is given by x=Σ_(q=0) ^(∞)b_(q)a^(q)[cos(φ)]^(q)  (61) Using the ratio

$\begin{matrix} {{\cos(\beta)} = {\sum\limits_{r = 0}^{{int}{({q/2})}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {2\; r} \end{pmatrix}\left( {\cos\;\beta} \right)^{q - {2\; r}}\left( {\sin\;\beta} \right)^{2\; r}}}} & (62) \end{matrix}$ which applies for any angle β, and the ratio (cos β)²+(sin β)²=1 which is also generally valid, it is however possible to determine coefficients g_(q,r) such that [cos(φ)]^(q=Σ) _(r=0) ^(q) g _(q,r) cos(rφ)  (63)

The coefficients g_(q,r) are independent of β or φ, as applicable. The first coefficients g_(q,r) turn out as g_(0,0)=1, g_(1,0)=0, g_(1,1)=1, g_(2,0)=½, g_(2,1)=0, g_(2,2)=−½, g_(3,0)=0, g_(3,1)=¾g_(3,2)=0, g_(3,3)=¼. This allows equation 61 to be rewritten as x=Σ _(q=0) ^(∞) b _(q) a ^(q)Σ_(r=0) ^(q) g _(q,r) cos(rφ)=Σ_(q=0) ^(∞) c _(q) cos(qφ)  (64) where c_(q)=Σ_(q′=q) ^(∞)b_(q′)a^(q′)g_(q′,q)  (65)

In practice, it is only necessary to consider a finite number of the coefficients bq. The others can to a good approximation be assumed to be zero. As a result, the system of equations in equation 65 is reduced to a finite system of equations, which for known correction values cq can be solved for the coefficients bq. The trigger produces a system of equations in the form of equation 60. The matrix coefficients hq,q′ can thus be determined by a comparison of coefficients. In this way one obtains, for example, h0,0=1, h0,1=0, h1,1=1, h0,2=−1, h1,2=0, h2,2=2, h0,3=0, h1,3=−3, h2,3=0, h3,3=4.

For the purpose of compensating for the errors in the measuring signals x, y, arising from the non-linear function f, one can simply subject the pre-corrected signals x_(c), y_(c) to an inverse mapping. For small errors, that is to say for |c_(q)|<<1, this inverse mapping is given approximately by x _(cc) =x _(c)−Σ_(q=2) ^(∞) b _(q) x _(c) ^(q)  (66) y _(cc) =y _(c)−Σ_(q=2) ^(∞) b _(q) y _(c) ^(q)  (67)

The above assumes throughout that the correction values x₀, y₀, m, Δ, c_(q), d_(q) are known, and thus it is possible to effect the compensation. However, the correction values x₀, y₀, m, Δ, c_(q), d_(q) must also be determined. For this purpose, we proceed according to FIG. 1, as follows:

For each position φ_(cc) which is determined, the sum of the squares of the corrected signals x_(cc), y_(cc), or the square root of this sum, is also determined as applicable. That is, from the corrected signals x_(cc), y_(cc) is derived a supplementary signal r_(cc) ² or r_(cc) as applicable, in the form r _(cc) ² =x _(cc) ² +y _(cc) ² and r _(cc)=√{square root over (x _(cc) ² +y _(cc) ²)}  (68)

In what follows, only the approach for a supplementary signal r_(cc) is considered. The approach for the supplementary signal r_(cc) ² is completely analogous.

The supplementary signal r_(cc) and the position φ_(cc) are fed into a Fourier block 8—see FIGS. 1 and 8. As shown in FIG. 8, the Fourier block 8 has a number of registers 9. The supplementary signal r_(cc) which is instantaneously being fed in is saved into one of these registers 9.

As shown in FIG. 9, an angular range, α1 to αn, is assigned to each of the registers 9, where n is preferably a power of 2. The Fourier block 9 then has a selector 10. The position signal φ_(cc) is fed to the selector 10. By reference to the position signal φ_(cc), the selector 10 activates that register 9 for which the position signal φ_(cc) lies within its assigned angular range α1 to αn, in order to save the supplementary signal r_(cc) concerned into this register 9.

In addition, a flag 11 is assigned to each register 9. As well as saving away the supplementary signal r_(cc) into one of the registers 9, the selector 10 at the same time also sets the flag 11 which is assigned to the register 9 concerned.

The flags 11 are linked to a trigger element 12. By reference to the flags 11, the trigger element 12 determines whether a trigger condition is satisfied. If the trigger condition is not satisfied, the trigger element 12 does not activate a computational block 13. On the other hand, if the trigger condition is satisfied, it activates the computational block 13. So a determination of the Fourier coefficients E_(i), F_(i) is only undertaken if the trigger condition is satisfied.

If the trigger condition is satisfied, the computational block 13 determines the Fourier coefficients E_(i), F_(i) for the supplementary signal r_(cc) by reference to the totality of the values saved in the registers 9. It thus determines the Fourier coefficients E_(i), F_(i) in such a way that the following applies r _(cc) =E ₀+Σ_(i=1) ^(∞) [E _(i) cos(iφ)+F _(i) sin(iφ)]  (69)

After the determination of the Fourier coefficients E_(i), F_(i), the computational block 13 resets the flags 11 again. Furthermore, it also clears the values saved in the registers 9. A re-determination of the Fourier coefficients E_(i), F_(i) will thus not take place again until the trigger condition is again satisfied.

In the simplest case, the trigger condition is only satisfied when values have been saved into all the registers 9 in accordance with the method described above. In this case, it is only necessary to check whether all the flags 11 have been set.

However, it is also possible for the trigger condition to be satisfied when values have been saved into only a first group of the registers 9 in accordance with the method described above. For example, it may be assumed that there is an adequate filling of the registers 9 if it is true for each register 9 that its assigned flag is set and/or the flags 11 assigned to both the immediately neighboring registers 9 are set. This can be determined—for each register 9 individually—by means of a logical combination, an example of which is shown in FIG. 10. In particular, in this case the remaining registers 9 can be filled with values which are determined by reference to the values already saved. For example, into each register 9, in which a value has not yet been saved in accordance with the above method, could be saved the mean of the two values which have been saved into the two registers 9 which are immediately neighboring in terms of angle.

The computational block 13 thus determines—see FIGS. 11 and 12—the Fourier coefficients E_(i)(i=0, 1, . . . ) and F_(i) (i=1, 2, . . . ) in a manner known per se. In principle, the Fourier coefficients E_(i), F_(i) are thus determined in the computational block 13 in accordance with the usual approach. For example, they can be determined in accordance with the formulae E ₀=(1/n)Σ_(m=0) ^(n−1) r _(cc)(m)  (70) E _(i)=[(1/(2n)]Σ_(m=0) ^(n−1) r _(cc)(m)cos(2πim/n)  (71) F _(i)=[(1/(2n)]Σ_(m=0) ^(n−1) r _(cc)(m)sin(2πim/n)  (72)

Preferably, however, certain registers 9 are assigned to each of the Fourier coefficients E_(i), F_(i). These registers 9 can, in particular, be those of the registers 9 for which the contribution, of the value saved in the register 9 concerned to the Fourier coefficients E_(i), F_(i) concerned, is particularly heavily weighted, i.e. the value of cos(2πim/n) or sin(2πim/n) lies close to one. The computational effort can then be significantly reduced without any essential change in the value determined for the Fourier coefficients E_(i), F_(i). It is thus possible to determine the Fourier coefficients E_(i), F_(i) concerned exclusively by reference to the values which are saved in the registers 9 assigned to the Fourier coefficients E_(i), F_(i). The registers 9 which are assigned to the Fourier coefficients E_(i), F_(i) concerned are here obviously determined individually for each Fourier coefficient E_(i), F_(i).

The approach just outlined can even be extended to the point that the only registers 9 assigned to each Fourier coefficient E_(i), F_(i) are those for which the cosine or sine, as applicable, assumes the maximum absolute value. In this case it is possible to determine the Fourier coefficients E_(i), F_(i) exclusively by the formation of sums and differences of the values saved in the assigned registers 9.

As can be seen from FIGS. 11 and 12, the offset correction values x₀, y₀ are determined from the Fourier coefficients E₁, F₁ for the basic frequency component of the supplementary signal r_(cc). The amplitude correction value m and the phase correction value Δ are determined from the Fourier coefficients E₂, F₂ for the first harmonic component in the supplementary signal r_(cc). Because, for small error variables x₀, y₀, m, Δ the following applies to a very good approximation E₀=a  (73) E ₁ =x ₀+(a/2)c ₂−(a/2)d ₂  (74) F ₁ =y ₀+(a/2)c ₂+(a/2)d ₂  (75) E ₂=−(a/2)m  (76) F ₂=−(a/2)Δ  (77)

Under the realistic assumption that the correction values c₂, d₂ vanish or are negligibly small relative to the offsets x₀, y₀, these equations thus give uniquely the base correction values (i.e. the offset, amplitude and phase correction values) x₀, y₀, m and Δ.

On the other hand, for higher-frequency waves in the measuring signals x, y, the assignment of the Fourier coefficients E_(i), F_(i) to the correction values c_(q), d_(q) are ambiguous. Because for n=0, 1, 2, . . . it is approximately true that E _(3+4n)=(a/2)(c _(2+4n) +d _(2+4n) +c _(4+4n) +d _(4+4n))  (78) F _(3+4n)=(a/2)(−c _(2+4n) +d _(2+4n) −c _(4+4n) +d _(4+4n))  (79) E _(4+4n) =a(c _(3+4n) +c _(5+4n))  (80) F _(4+4n) =a(d _(3+4n) +d _(5+4n))  (81) E _(5+4n)=(a/2)(c _(4+4n) −d _(4+4n) +c _(6+4n) −d _(6+4n))  (82) F _(5+4n)=(a/2)(c _(4+4n) +d _(4+4n) +c _(6+4n) +d _(6+4n))  (83)

These ambiguities can be resolved in various ways.

Namely, the above system of equations has been derived using first partial derivatives. It is therefore possible, for example, to take into account also higher order derivatives, and thus to arrive at further ratios between the Fourier coefficients E_(i), F_(i) on the one hand, and the correction values c_(q), d_(q) on the other hand. The ambiguity could possibly be eliminated in this way. However, this approach requires a very high computational effort. Also, the resulting system of equations is generally no longer analytically soluble, but only numerically.

In practice, however, one can often make simplifying assumptions, on the basis of which the assignment of the Fourier coefficients E_(i), F_(i) to the correction values c_(q), d_(q) becomes unambiguous.

A first possible assumption is that in the measuring signals x, y the higher-frequency waves which arise are essentially only those with a frequency which is an odd number multiple of the basic frequency fG. The result of this assumption, which is most cases is perfectly applicable, is that only equations 80 and 81 must be solved. So, it is only necessary to determine the correction values c_(q), d_(q) for the at least one higher-frequency waves, in the measuring signals x, y, the frequency of which is an odd integral multiple of the basic frequency fG of the corrected signals x_(cc), y_(cc).

It can also be assumed without major error that the only relevant higher-frequency waves in the measuring signal x,y are those with a frequency of three, and possibly also five times the basic frequency fG of the corrected signals x_(cc), y_(cc). It is therefore sufficient to solve equations 80 and 81 for n=0, and thus to determine the correction values c₃, d₃, c₅, d₅. For this purpose there are two alternative possibilities, which are shown in FIGS. 11 and 12.

On the one hand it can be assumed—see FIG. 11—that the correction values c₅, d₅ have a predetermined ratio to the correction values c₃, d₃. For example, it is assumed that the correction values c₃ and c₅ are in the ratio 3:1, that is the correction value c₃ is always three times as large as the correction value c₅. Other (even negative) ratios are however also conceivable. With this assumption, the correction values c₃ and c₅ can be determined uniquely from equation 80. For the correction values d₃ and d₅, either the same assumption can be made or a different one.

Alternatively it can also be assumed—so to speak as a special case of this approach—that the harmonic wave with a frequency five times that of the basic frequency fG of the corrected signals x_(cc), y_(cc) vanishes, that is the correction values c₅ and d₅ have a value of zero. In this case it is only necessary to determine the correction values c₃, d₃ for those higher-frequency waves in the measuring signals x, y with a frequency which is three times the basic frequency fG. Then in this case, for example, c₃=E₄/E₀. This approach is shown in FIG. 12.

Depending on the situation, it can indeed be logical in an individual case to assume that both the correction values d₃ and d₅ and also the correction value c₅ vanish, i.e. have a value of zero.

Using the Fourier coefficients E_(i), F_(i) it is then possible to adjust the correction values x₀, y₀, m, Δ, c_(q), d_(q). For example, in the case where correction values c₃, d₃ are determined only for the third harmonic, the following adjustment rules can be executed: a:=a+αE ₀  (84) x ₀ :=x ₀ +αE ₁  (85) y ₀ :=y ₀ +αF ₁  (86) m:=m−2αE ₂ /E ₀  (87) Δ:=Δ−2αF ₂ /E ₀  (88) c ₃ :=c ₃ +αE ₄ /E ₀  (89) d ₃ :=d ₃ +αF ₄ /E ₀  (90)

Here, the factor α is a positive number which is less than one. It is preferably the same for all the adjusted values a, x₀, y₀, m, Δ, c₃, d₃. However, it can also be defined separately for each individual value which is adjusted, a, x₀, y₀, m, Δ, c₃, d₃.

The above is a description of the fact that, and how, the correction values x₀, y₀, m, Δ, c₃, d₃ have been determined using a supplementary signal r_(cc). Here, the supplementary signal r_(cc) (or r_(cc) ² as applicable) corresponded respectively to the sum of the squares of the corrected signals x_(cc), y_(cc), or the square root of this sum.

By means of the approach in accordance with the invention it is thus also possible to correct higher-frequency waves in the measuring signals x, y in a simple manner. This is indicated in FIG. 1 by dashed lines. In this case, the equations x _(cc) =x ₀ +a cos(φ_(cc)+Δ)+aΣ _(q=2) ^(∞) [c _(q) cos(qφ _(cc) +qΔ)+d _(q) sin(qφ _(cc) +qΔ)]  (91) y _(cc) =y ₀ +a(1+m)sin(φ_(cc))+a(1+m)Σ_(q=2) ^(∞) [c _(q) cos(qφ _(cc) −qπ/2)+d _(q) sin(qφ _(cc) +qπ/2)]  (92) must be equated to the corresponding Fourier expansions x _(cc) =XR ₀+Σ_(q=1) ^(∞) [XR _(q) cos(qφ _(cc))+XI _(q) sin(qφ _(cc))  (93) y _(cc) =YR ₀+Σ_(q=1) ^(∞) [YR _(q) cos(qφ _(cc))+YI _(q) sin(qφ _(cc))  (94)

In this case, the assignment of the Fourier coefficients XR_(q), XI_(q), YR_(q), YI_(q) to the correction values c_(q), d_(q) can be made simply and uniquely. However, the principle of the approach, that is in particular the manner in which the Fourier coefficients XR_(q), XI_(q), YR_(q), YI_(q) are determined, the adjustment of the correction values x₀, y₀, m, Δ, c_(q) and d_(q) by reference to the Fourier coefficients XR_(q), XI_(q), YR_(q), YI_(q) which have been determined, and the determination of the corrected signals x_(cc), y_(cc) by reference to the measuring signals x, y and the correction values x₀, y₀, m, Δ, c_(q), d_(q), is just as previously described for the supplementary signal r_(cc).

In particular cases, there may be small differences between the correction values c_(q), d_(q) determined by evaluation of the equations 91 and 93 on the one hand and 92 and 94 on the other. For this reason it is preferable, as shown in FIG. 13, to determine the Fourier coefficients XR_(q), XI_(q), YR_(q), YI_(q) for both corrected signals x_(cc), y_(cc). In this case, the correction values c_(q), d_(q) for the higher-frequency waves in the measuring signals x, y will be adjusted using the Fourier coefficients XR_(q), XI_(q), YR_(q), YI_(q) for both corrected signals x_(cc), y_(cc). In particular, mean values can be formed.

Unlike the sums of the squares of the corrected signals x_(cc), y_(cc), the corrected signals x_(cc), y_(cc) themselves show a marked fluctuation at the basic frequency fG. It can be logical therefore to begin by using the arctangent φ_(cc) and the amplitude a to determine expected signals x′, y′ according to the equations x′=a cos φ_(cc) and  (95) y′=a sin φ_(cc)  (96) and to subtract these expected signals x′, y′ from the corresponding measuring signals x, y. That is to say, in this case supplementary signals δx, δy are formed, corresponding to the difference between the measuring signals x, y and the expected signals x′, y′. The correction values x₀, y₀, m, Δ, c_(q), d_(q) are in this case adjusted using the Fourier coefficients of the supplementary signals δx, δy.

By means of the approach in accordance with the invention it is thus also possible to correct higher-frequency waves in the measuring signals x, y in a simple manner. 

1. A determination method for correcting errors in a position signal, comprising: scanning a measuring scale having a plurality of equally-spaced scale divisions by at least two sensors which are moveable relative to the measuring scale; supplying measuring signals based upon the scanning, wherein the measuring signals: are periodic for a uniform relative movement of the measuring scale, have essentially the same amplitude, are essentially sinusoidal, have a phase offset relative to one another which is essentially 90°, and have a basic frequency which corresponds with the relative movement of the measuring scale, and during the course of one period of the measuring signals, the measuring scale executes a relative movement of one scale division; determining corrected signals from the measuring signals based upon correction values; determining a position signal of the measuring scale relative to the sensors based upon the corrected signals; determining Fourier coefficients for a supplementary signal based upon the basic frequency, wherein the supplementary signal is equal to a sum of squares of the corrected signals or a value derived from this sum; and adjusting the correction values based upon the Fourier coefficients; wherein the correction values are based upon a value selected from the group consisting of: at least two offset correction values, at least one amplitude correction value, at least one phase correction value for the measuring signals and combinations thereof, together with at least one correction value for at least one higher-frequency wave in the measuring signals.
 2. The determination method as claimed in claim 1, wherein the correction values for the at least one higher-frequency wave in the measuring signal is determined only for higher-frequency waves in the measuring signal which have a frequency which is an odd number multiple of the basic frequency.
 3. The determination method as claimed in claim 2, wherein the correction values are determined only for the higher-frequency waves in the measuring signal, the higher-frequency being three or five times the basic frequency, wherein further the correction values for the higher-frequency waves in the measuring signals which have a frequency of five times the basic frequency have a predetermined ratio to the correction values for the higher-frequency waves in the measuring signals, which have a frequency three times the basic frequency.
 4. The determination method as claimed in claim 3, wherein the correction values are determined only for the higher-frequency waves in the measuring signal which have a frequency which is three times the basic frequency.
 5. The determination method as claimed in claim 1, further comprising: saving the supplementary signal in at least one register which has an assigned angular range, using the saved supplementary signal to determine the Fourier coefficients, saving the supplementary signal in the register whose angular range lies in the arctangent of the corrected signals, and determining the Fourier coefficients based upon the values saved into the registers.
 6. The determination method as claimed in claim 1, wherein a plurality of registers and an angular range is assigned to each register.
 7. The determination method as claimed in claim 6, wherein after the determination of the Fourier coefficients the values saved into the registers are deleted and a re-determination of the Fourier coefficients will only be undertaken again after the registers have been filled sufficiently.
 8. The determination method as claimed in claim 7, wherein the registers are filled when values have been saved into a first group of the registers, and a second group of the registers are filled with values which are determined by reference to the values saved.
 9. The determination method as claimed in claim 5, wherein registers are assigned to each Fourier coefficient and the Fourier coefficient is determined only based upon the values which are saved in the registers assigned to the Fourier coefficient.
 10. The determination method as claimed in claim 9, wherein the Fourier coefficients are determined by forming sums and differences of the values saved in the assigned registers.
 11. The determination method as claimed in claim 1, further comprising: determining pre-corrected signals determining pre-corrected signals based upon: the measuring signals, and a value selected from the group consisting of: the offset correction value, the at least one amplitude correction value, the at least one phase correction value, and combinations thereof determining the corrected signals based upon the pre-corrected signals and to the at least one correction value for the one or more higher-frequency waves in the measuring signals.
 12. The determination method as claimed in claim 11, further comprising: determining a preliminary arctangent based upon the pre-corrected signals, and determining the corrected signals applying the preliminary arctangent as the argument in a Fourier series expansion.
 13. The determination method as claimed in claim 12, wherein the corrected signals are determined based upon the pre-corrected signals by the formation of functions of the form $\begin{matrix} {x_{cc} = {x_{c} - {a\;{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}{\cos\left( {q\;\varphi_{c}} \right)}} + {d_{q}{\sin\left( {q\;\varphi_{c}} \right)}}} \right\rbrack}}}} & \; \\ {and} & \; \\ {y_{cc} = {y_{c} - {a\;{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}{\cos\left( {{q\;\varphi_{c}} - {q\;{\pi/2}}} \right)}} + {d_{q}{\sin\left( {{q\;\varphi_{c}} - {q\;{\pi/2}}} \right)}}} \right\rbrack}}}} & \; \end{matrix}$ where: x_(cc) and y_(cc) are the corrected signals, x_(c) and y_(c) are the pre-corrected signals, a is the signal amplitude, c_(q) and d_(q) are weighting factors determined based upon the Fourier coefficients, and φ_(c) is the preliminary arctangent.
 14. The determination method as claimed in claim 13, wherein in the formula $y_{{cc}\;} = {y_{c} - {a{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}{\cos\left( {{q\;\varphi_{c}} - {q\;{\pi/2}}} \right)}} + {d_{q}{\sin\left( {{q\;\varphi_{c}} - {q\;{\pi/2}}} \right)}}} \right\rbrack}}}$ the expression cos(qφ_(c)−qπ/2) is replaced by cos(qφ_(c)) for q=0, 4, 8, . . . , by sin(qφ_(c)) for q=1, 5, 9, . . . , by −cos(qφ _(c)) for q=2, 6, 10, . . . , and by −sin(qφ _(c)) for q=3, 7, 11, . . . . and the expression sin(qφ_(c)−qπ/2) is replaced by sin(qφ_(c)) for q=0, 4, 8, . . . , by −cos(qφ _(c)) for q=1, 5, 9, . . . , by −sin(qφ _(c)) for q=2, 6, 10, . . . , and by cos(qφ_(c)) for q=3, 7, 11, . . . .
 15. The determination method as claimed in claim 12, wherein the corrected signals are determined based upon the pre-corrected signals by the formation of functions of the form $x_{cc} = {x_{c} - {a{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}z_{3}} + {d_{q}z_{4}}} \right\rbrack}}}$ and $y_{cc} = {y_{c} - {a\;{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}z_{1}} + {d_{q}z_{2}}} \right\rbrack}}}$ where: x_(cc) and y_(cc) are the corrected signals, x_(c) and y_(c) are the pre-corrected signals, a is the signal amplitude, c_(q) and d_(q) are weighting factors determined based upon the Fourier coefficients, φ_(c) is the preliminary arctangent z₁ is z₃ for q=0, 4, 8, . . . , z₄ for q=1, 5, 9, . . . , −z₃ for q=2, 6, 10, . . . , and −z₄ for q=3, 7, 11, . . . , z₂ is z₄  for  q = 0, 4, 8, …  , −z₃  for  q = 1, 5, 9, …  , −z₄  for  q = 2, 6, 10, …  , and ${{z_{3}\mspace{14mu}{for}\mspace{14mu} q} = 3},7,11,\ldots\mspace{11mu},{z_{3} = {\sum\limits_{r = 0}^{{int}{({q/2})}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {2r} \end{pmatrix}\left( {\cos\;\varphi_{c}} \right)^{q - {2\; r}}\left( {\sin\;\varphi_{c}} \right)^{2\; r}}}},{and}$ $z_{4} = {\sum\limits_{r = 0}^{{int}{\lbrack{{({q - 1})}/2}\rbrack}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {{2r} + 1} \end{pmatrix}\left( {\cos\;\varphi_{c}} \right)^{q - {2r} - 1}{\left( {\sin\;\varphi_{c}} \right)^{{2r} + 1}.}}}$
 16. The determination method as claimed in claim 12, wherein the corrected signals are determined based upon the pre-corrected signals by the formation of functions of the form $\left. {{x_{cc} = {x_{c} - {a{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}z_{3}} + {d_{q}z_{4}}} \right\rbrack}}}}{and}{y_{cc} = {y_{c} - {a{\sum\limits_{q = 2}^{\infty}\left\lbrack {{c_{q}z_{1}} + {d_{q}z_{2}}} \right)}}}}} \right\rbrack$ where: x_(cc) and y_(cc) are the corrected signals, x_(c) and y_(c) are the pre-corrected signals, a is the signal amplitude, c_(q) and d_(q) are weighting factors determined based upon the Fourier coefficients, φ_(c) is the preliminary arctangent z₁ is z₃ for q=0, 4, 8, . . . , z₄ for q=1, 5, 9, . . . , −z₃ for q=2, 6, 10, . . . , and −z₄ for q=3, 7, 11, . . . , z₂ is z₄  for  q = 0, 4, 8, …  , −z₃  for  q = 1, 5, 9, …  , −z₄  for  q = 2, 6, 10, …  , and ${{z_{3}\mspace{14mu}{for}\mspace{14mu} q} = 3},7,11,\ldots\mspace{11mu},{z_{3} = {\sum\limits_{r = 0}^{{int}{({q/2})}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {2r} \end{pmatrix}\left( {x\;{c/a}} \right)^{q - {2\; r}}\left( {y\;{c/a}} \right)^{2\; r}}}},{and}$ $z_{4} = {\sum\limits_{r = 0}^{{int}{\lbrack{{({q - 1})}/2}\rbrack}}{\left( {- 1} \right)^{r}\begin{pmatrix} q \\ {{2r} + 1} \end{pmatrix}\left( {x\;{c/a}} \right)^{q - {2r} - 1}{\left( {y\;{c/a}} \right)^{{2r} + 1}.}}}$
 17. The determination method as claimed in claim 11, wherein the corrected signals are determined based upon the pre-corrected signals, by the formation of functions of the form $x_{cc} = {x_{c} - {\sum\limits_{q = 2}^{\infty}{b_{q}x_{c}^{q}}}}$ and $y_{cc} = {y_{c} - {\sum\limits_{q = 2}^{\infty}{b_{q}y_{c}^{q}}}}$ where x_(cc) and y_(cc) are the corrected signals and x_(c) and y_(c) the pre-corrected signals, and b_(q) a weighting factor. 